Lecture 18
Community structures
Slides modified from Huan Liu, Lei Tang, Nitin Agarwal
Community Detection
 A community is a set of nodes between which the
interactions are (relatively) frequent
a.k.a. group, subgroup, module, cluster
 Community detection
a.k.a. grouping, clustering, finding cohesive subgroups
 Given: a social network
 Output: community membership of (some) actors
 Applications
 Understanding the interactions between people
 Visualizing and navigating huge networks
 Forming the basis for other tasks such as data mining
2
Visualization after Grouping
4 Groups:
{1,2,3,5}
{4,8,10,12}
{6,7,11}
{9,13}
(Nodes colored by
Community Membership)
3
Classification
 User Preference or Behavior can be represented as
class labels
• Whether or not clicking on an ad
• Whether or not interested in certain topics
• Subscribed to certain political views
• Like/Dislike a product
 Given
 A social network
 Labels of some actors in the network
 Output
 Labels of remaining actors in the network
4
Visualization after Prediction
: Smoking
: Non-Smoking
: ? Unknown
Predictions
6: Non-Smoking
7: Non-Smoking
8: Smoking
9: Non-Smoking
10: Smoking
5
Link Prediction
 Given a social network, predict which nodes are likely to
get connected
 Output a list of (ranked) pairs of nodes
 Example: Friend recommendation in Facebook
(2, 3)
(4, 12)
(5, 7)
(7, 13)
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Viral Marketing/Outbreak Detection
 Users have different social capital (or network values)
within a social network, hence, how can one make best
use of this information?
 Viral Marketing: find out a set of users to provide
coupons and promotions to influence other people in the
network so my benefit is maximized
 Outbreak Detection: monitor a set of nodes that can help
detect outbreaks or interrupt the infection spreading
(e.g., H1N1 flu)
 Goal: given a limited budget, how to maximize the
overall benefit?
7
An Example of Viral Marketing
 Find the coverage of the whole network of nodes with
the minimum number of nodes
 How to realize it – an example
 Basic Greedy Selection: Select the node that maximizes the
utility, remove the node and then repeat
• Select Node 1
• Select Node 8
• Select Node 7
Node 7 is not a node with
high centrality!
8
PRINCIPLES OF
COMMUNITY DETECTION
Communities
 Community: “subsets of actors among whom there are
relatively strong, direct, intense, frequent or positive
ties.”
-- Wasserman and Faust, Social Network Analysis, Methods and Applications
 Community is a set of actors interacting with each other
frequently
 A set of people without interaction is NOT a community
 e.g. people waiting for a bus at station but don’t talk to each
other
10
Example of Communities
Communities from
Facebook
Communities from
Flickr
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Community Detection
 Community Detection: “formalize the strong social
groups based on the social network properties”
 Some social media sites allow people to join groups
 Not all sites provide community platform
 Not all people join groups
 Network interaction provides rich information about the
relationship between users
 Is it necessary to extract groups based on network topology?
 Groups are implicitly formed
 Can complement other kinds of information
 Provide basic information for other tasks
12
Subjectivity of Community Definition
A densely-knit
community
Each component is
a community
Definition of a community
can be subjective.
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Taxonomy of Community Criteria
 Criteria vary depending on the tasks
 Roughly, community detection methods can be divided
into 4 categories (not exclusive):
 Node-Centric Community
 Each node in a group satisfies certain properties
 Group-Centric Community
 Consider the connections within a group as a whole. The group
has to satisfy certain properties without zooming into node-level
 Network-Centric Community
 Partition the whole network into several disjoint sets
 Hierarchy-Centric Community
 Construct a hierarchical structure of communities
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Node-Centric Community Detection
NodeCentric
HierarchyCentric
Community
Detection
NetworkCentric
GroupCentric
Node-Centric Community Detection
 Nodes satisfy different properties
 Complete Mutuality
 cliques
 Reachability of members
 k-clique, k-clan, k-club
 Nodal degrees
 k-plex, k-core
 Relative frequency of Within-Outside Ties
 LS sets, Lambda sets
 Commonly used in traditional social network analysis
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Complete Mutuality: Clique
 A maximal complete subgraph of three or more nodes all
of which are adjacent to each other

NP-hard to find the maximal clique

Recursive pruning: To find a clique
of size k, remove those nodes with
less than k-1 degrees

Normally use cliques as a core or
seed to explore larger communities
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Geodesic
 Reachability is calibrated by the
Geodesic distance
 Geodesic: a shortest path between
two nodes (12 and 6)
 Two paths: 12-4-1-2-5-6, 12-10-6
 12-10-6 is a geodesic
 Geodesic distance: #hops in geodesic
between two nodes
 e.g., d(12, 6) = 2, d(3, 11)=5
 Diameter: the maximal geodesic
distance for any 2 nodes in a network
Diameter = 5
 #hops of the longest shortest path
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Reachability: k-clique, k-club
 Any node in a group should be
reachable in k hops
 k-clique: a maximal subgraph in which
the largest geodesic distance between
any nodes <= k
 A k-clique can have diameter larger
than k within the subgraph
 e.g., 2-clique {12, 4, 10, 1, 6}
 Within the subgraph d(1, 6) = 3
 k-club: a substructure of diameter <= k
 e.g., {1,2,5,6,8,9}, {12, 4, 10, 1} are 2-clubs
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Nodal Degrees: k-core, k-plex
 Each node should have a certain number of connections
to nodes within the group
 k-core: a substracture that each node connects to at
least k members within the group
 k-plex: for a group with ns nodes, each node should
be adjacent no fewer than ns-k in the group
 The definitions are complementary
 A k-core is a (ns-k)-plex
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Within-Outside Ties: LS sets
 LS sets: Any of its proper subsets has more ties to other
nodes in the group than outside the group
 Too strict, not reasonable for network analysis
 A relaxed definition is Lambda sets
 Require the computation of edge-connectivity between any
pair of nodes via minimum-cut, maximum-flow algorithm
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Recap of Node-Centric Communities
 Each node has to satisfy certain properties
 Complete mutuality
 Reachability
 Nodal degrees
 Within-Outside Ties
 Limitations:
 Too strict, but can be used as the core of a community
 Not scalable, commonly used in network analysis with small-size
network
 Sometimes not consistent with property of large-scale networks
 e.g., nodal degrees for scale-free networks
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Group-Centric Community Detection
NodeCentric
HierarchyCentric
Community
Detection
NetworkCentric
GroupCentric
Group-Centric Community Detection
 Consider the connections within a group as whole,
 Some nodes may have low connectivity
 A subgraph with Vs nodes and Es edges is a γ-dense
quasi-clique if
 Recursive pruning:
 Sample a subgraph, find a maximal γ-dense quasi-clique
 the resultant size = k
 Remove the nodes that
 whose degree < kγ
 all their neighbors with degree < kγ
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Network-Centric Community Detection
NodeCentric
HierarchyCentric
Community
Detection
NetworkCentric
GroupCentric
Network-Centric Community Detection
 To form a group, we need to consider the connections of
the nodes globally.
 Goal: partition the network into disjoint sets
 Groups based on
 Node Similarity
 Latent Space Model
 Block Model Approximation
 Cut Minimization
 Modularity Maximization
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Node Similarity
 Node similarity is defined by how similar their interaction
patterns are
 Two nodes are structurally equivalent if they connect to
the same set of actors
 e.g., nodes 8 and 9 are structurally equivalent
 Groups are defined over equivalent nodes
 Too strict
 Rarely occur in a large-scale
 Relaxed equivalence class is difficult to compute
 In practice, use vector similarity
 e.g., cosine similarity, Jaccard similarity
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Vector Similarity
1
2
3
4
1
5
6
7
8
9
10
11 12 13
a vector
5
1
structurally
equivalent
8
1
1
1
9
1
1
1
Cosine Similarity:
sim (5,8) 
1
1

2 3
6
Jaccard Similarity:
J (5,8)  |{1, 2|{,66},|13}|  1 / 4
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Clustering based on Node Similarity
 For practical use with huge networks:
 Consider the connections as features
 Use Cosine or Jaccard similarity to compute vertex similarity
 Apply classical k-means clustering Algorithm
 K-means Clustering Algorithm
 Each cluster is associated with a centroid (center point)
 Each node is assigned to the cluster with the closest centroid
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Illustration of k-means clustering
Iteration 1
Iteration 2
Iteration 3
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Groups on Latent-Space Models
 Latent-space models: Transform the nodes in a network into a
lower-dimensional space such that the distance or similarity between
nodes are kept in the Euclidean space
 Multidimensional Scaling (MDS)
 Given a network, construct a proximity matrix to denote the distance between
nodes (e.g. geodesic distance)
 Let D denotes the square distance between nodes
 S  R nk denotes the coordinates in the lower-dimensional space
1
1
1
SS T   ( I  eeT ) D( I  eeT )  ( D)
2
n
n
 Objective: minimize the difference
 Let
min || ( D)  SS T || F
(the top-k eigenvalues of
), V the top-k eigenvectors
 Solution:
 Apply k-means to S to obtain clusters
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MDS-example
1, 2, 3, 4,
10, 12
5, 6, 7, 8,
9, 11, 13
k-means
Geodesic Distance Matrix
1
2
3
4
5
6
7
8
9
10
1
0
1
1
1
2
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3
1
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2
9 10 11 12 13
1 2 4 2 2
2 3 4 3 3
2 3 5 3 3
2 1 4 1 3
2 2 3 3 3
1 1 2 2 2
2 2 1 3 3
2 2 3 3 1
0 2 3 3 1
2 0 3 1 3
4
1
3
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MDS
S
-1.22
-0.88
-2.12
-1.01
0.43
0.78
1.81
-0.09
-0.09
0.30
2.85
-0.47
-0.29
-0.12
-0.39
-0.29
1.07
-0.28
0.04
0.02
-0.77
-0.77
1.18
0.00
2.13
-1.81
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Block-Model Approximation
After
Reordering
Network Interaction Matrix
Block Structure
Objective: Minimize the difference between an interaction
matrix and a block structure
S is a
community
indicator matrix
Challenge: S is discrete, difficult to solve
Relaxation: Allow S to be continuous satisfying
Solution: the top eigenvectors of A
Post-Processing: Apply k-means to S to find the partition
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Cut-Minimization
 Between-group interactions should be infrequent
 Cut: number of edges between two sets of nodes
 Objective: minimize the cut
 Limitations: often find communities of
only one node
 Need to consider the group size
 Two commonly-used variants:
Cut=2
Number of nodes
in a community
Cut =1
Number of
within-group
Interactions
36
Modularity Maximization
 Modularity measures the group interactions compared
with the expected random connections in the group
 In a network with m edges, for two nodes with degree di
and dj , expected random connections between them are
 The interaction utility in a group:
 To partition the group into
multiple groups, we maximize
Expected Number of
edges between 6 and 9
is
5*3/(2*17) = 15/34
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Properties of Modularity
 Properties of modularity:
 Between (-1, 1)
 Modularity = 0 If all nodes are clustered into one group
 Can automatically determine optimal number of clusters
 Resolution limit of modularity
 Modularity maximization might return a community consisting
multiple small modules
40
Recap of Network-Centric Community
 Network-Centric Community Detection
 Groups based on
 Node Similarity
 Latent Space Models
 Cut Minimization
 Block-Model Approximation
 Modularity maximization
 Goal: Partition network nodes into several disjoint sets
 Limitation: Require the user to specify the number of
communities beforehand
42
Hierarchy-Centric Community Detection
NodeCentric
HierarchyCentric
Community
Detection
NetworkCentric
GroupCentric
Hierarchy-Centric Community Detection
 Goal: Build a hierarchical structure of communities
based on network topology
 Facilitate the analysis at different resolutions
 Representative Approaches:
 Divisive Hierarchical Clustering
 Agglomerative Hierarchical Clustering
44
Divisive Hierarchical Clustering
 Divisive Hierarchical Clustering
 Partition the nodes into several sets
 Each set is further partitioned into smaller sets
 Network-centric methods can be applied for partition
 One particular example is based on edge-betweenness
 Edge-Betweenness: Number of shortest paths between any pair of nodes that
pass through the edge
 Between-group edges tend to have larger edge-betweenness
45
Divisive clustering on Edge-Betweenness
 Progressively remove edges with the highest
3
betweenness
3
3
5
 Remove e(2,4), e(3, 5)
5
 Remove e(4,6), e(5,6)
 Remove e(1,2), e(2,3), e(3,1)
4
4
root
V1,v2,v3
v1
v2
V4, v5, v6
v3
v4
v5
v6
46
Agglomerative Hierarchical Clustering
 Initialize each node as a community
 Choose two communities satisfying certain criteria and
merge them into larger ones
 Maximum Modularity Increase
 Maximum Node Similarity
root
V4, v5, v6
V1, v2, v3
v3
V1,v2
v1
v2
V1,v2
v4
(Based on Jaccard Similarity)
v5
v6
47
Recap of Hierarchical Clustering
 Most hierarchical clustering algorithm output a binary
tree
 Each node has two children nodes
 Might be highly imbalanced
 Agglomerative clustering can be very sensitive to the
nodes processing order and merging criteria adopted.
 Divisive clustering is more stable, but generally more
computationally expensive
48
Summary of Community Detection
 The Optimal Method?
 It varies depending on applications, networks,
computational resources etc.
NodeCentric
Hierarchy
-Centric
 Other lines of research
 Communities in directed networks
 Overlapping communities
 Community evolution
 Group profiling and interpretation
Community
Detection
GroupCentric
NetworkCentric
49